Most wall damage falls into one of three categories. Once you know which type you're dealing with, the repair is straightforward.
Apply lightweight spackle with your finger or a 2-inch putty knife. Let dry. Sand lightly with 220-grit. Prime and paint.
Stick a self-adhesive mesh patch over the hole. Apply joint compound over the patch using a 6-inch knife. Feather the edges outward. Let dry, sand smooth, apply a second coat if needed. Prime and paint.
Cut the damage into a clean square. Install backing strips. Screw new drywall into the backing. Tape seams with mesh tape. Apply joint compound in coats. Sand between coats. Prime and paint.
If you skip primer, the patch will flash and look dull under paint. One coat of primer solves all of that.